基本quandle不决定2-纽结的第一个Postnikov不变量
Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots
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中文总结 AI 辅助
针对Plotnick--Suciu构造及Suciu的替代构造，研究发现基本quandle等对象无法决定定向2-纽结外部空间的同伦类型，还明确了Tanaka--Taniguchi例子的二阶同伦模性质。
中文摘要 AI 辅助
针对Plotnick--Suciu构造中每一对可允许的Brieskorn参数，我们得到一对定向2-纽结，其纽结群同构，在合适的群同构下其二阶同伦模半线性同构，且其基本quandle同构，但不存在兼容的群与模同构将其中一个的第一个Postnikov不变量映射到另一个的该不变量。因此，它们的外部空间不是同伦等价的。由此，纽结群、半线性等价下的二阶同伦模以及基本quandle都不决定定向2-纽结外部空间的同伦类型。利用Suciu论文中基于打孔透镜空间的替代构造，对于每个N≥2，同样的外围论证可得到具有上述性质的N个定向2-纽结族。我们还证明，具有同构纽结群和不同基本quandle的Tanaka--Taniguchi例子，其对应的二阶同伦模两两不等价：两个纽结群之间不存在同构使得对应的二阶同伦模半线性同构。
英文摘要
For every admissible pair of Brieskorn parameters in the Plotnick--Suciu construction, we obtain a pair of oriented $2$-knots whose knot groups are isomorphic, whose second homotopy modules are semilinearly isomorphic under a suitable group isomorphism, and whose fundamental quandles are isomorphic, while no compatible group and module isomorphisms carry one first Postnikov invariant to the other. Consequently, their exteriors are not homotopy equivalent. Thus, the knot group, the second homotopy module up to semilinear equivalence, and the fundamental quandle do not determine the homotopy type of an oriented $2$-knot exterior. Using an alternative construction from Suciu's thesis based on punctured lens spaces, the same peripheral argument yields, for every $N\geq2$, a family of $N$ oriented $2$-knots with these properties. We additionally show that the Tanaka--Taniguchi examples with isomorphic knot groups and distinct fundamental quandles have pairwise inequivalent second homotopy modules: no isomorphism between two of the knot groups makes the corresponding second homotopy modules semilinearly isomorphic.
